But computing the exact values of the codimensions of a given algebra is also a very difficult task, and \(c_n(A)\)  is known for very few algebras \(A\) . This is exactly the same list as above, namely that of the algebras whose identities are known. Hence one is led to study the growth of the codimension sequences. In the eighties Amitsur conjectured that for each PI-algebra \(A\) , the sequence \((c_n(A))^{1/n}\)  converges when \(n\rightarrow \infty \)  and moreover its limit is an integer. This conjecture was dealt with by Giambruno and Zaicev [1]}, [2]} (see also [3]}): they answered in the affirmative Amitsur's conjecture. The above limit is called the PI-exponent of a PI-algebra, \(\exp (A)\) . Giambruno and Zaicev's important result initiated an extensive research concerning the asymptotic behaviour of the codimension sequences of algebras. It is well known (see for example [3]} and the references therein) that either the codimensions of \(A\)  are bounded by a polynomial function or grow exponentially. The results of Giambruno and Zaicev also hold for the case of graded algebras ([5]}, [6]}, [7]}), algebras with involution ([8]}), superalgebras with superinvolution, graded involution or pseudoinvolution ([9]}, [10]}, [11]}) and also for large classes of non-associative algebras. It is useful to highlight that there are examples of non-associative algebras such that their PI-exponent exists but is not an integer and also examples where the PI-exponent does not exist at all.
